A simple card-flipping rule always seems to end — but nobody can prove it always does
Shuffle a deck of cards numbered 1 to n face up. Look at the card on top: if it shows the number k, flip over the top k cards as a single block, then repeat the process with whatever card is now on top. John Conway's Topswops asks whether this always has to stop, and if so, how long the longest possible game can last for a deck of a given size.
Reveal the answer
Every shuffle tested so far does eventually stop, and for small decks the maximum game lengths have been computed exactly, but whether the process is guaranteed to terminate for every deck size and every starting shuffle remains an open problem in general. John Conway posed Topswops (originally 'Topswaps') and offered a prize for progress on it, and Martin Gardner later popularized it in his Scientific American column.
— John Horton Conway, Topswops — Posed c. 1973; popularized by Martin Gardner in Scientific American